(1) Let (X, 7) be a topological space and let A CX. Let TA= {UNA: U ET}. (a) Show that TA is a topology on A. (b) Show that if 7 is Hausdorff, then T|A is Hausdorff.

Q: Q6: ) Suppose (X, 7) is aftopological space, and p a point not in X. Let Q = {UU {p}, UET, and X-U…

A: Let's go through the two parts of the question:a) To show that T=Q∪T is a topology on Y=X∪{p}, need…

Q: (1) Let X be an infinite set and set r to be the set given by T = {0} U {U C X : U° finite}. (a)…

A: τ={ϕ}∪{U⊂X :Uc is finite }  (a) Xc=ϕ, which is finite . so X∈τ let A,B∈τ Then Ac,Bcare finite…

Q: Q1:A) Let X = R and r be the usual topology on R.If Y-[4,6). 1) Determine the open ,closed ,open and…

A: Given that, X=Rand τ is usual topology on R Given that Y=[4,6). 1) 4,5 Is open in the given topology…

Q: Let (X,T) and (Y,T1) be two topological spaces and let f be a continuous mapping of X into Y. * None…

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Q: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and BC(X,M) := {f €…

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Q: For any topological space (X, T) and A C X, the set A is closed. That is, for any set A in a…

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: (3) Consider the topological space (R. 7), where T= {UCR: U = 0 or R\U is countable}. (a) Show that…

A: As we know τ is a topology on R if1. Both the empty set and R are elements of τ.2. Any union of…

Q: LxercisE Let (X,2,4) be a mensure spsce. Let 1 2- (0, 4oo and pn : 2(0,+o0] be two menaures. Prove…

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Q: Let (X, Tx) represent a topological space and {a} a singleton set, which is an open subset of X, {a}…

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Q: Let (X,T) be a separable topological space, let A be an open subset of X. Then (A, T,) is a…

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Q: 4) Let T be the Euclidean topology on R and T1 be the discrete topology on R. Define f : (R, 71) →…

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Q: Let X be a topological space

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Q: d T = {0, {a}, {a, b}, {c,d}, {a, c, d}, {a,b, c, d}, X}. (1) Is this space To? Why or why not? (2)…

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Q: cise 2 Let X be a topological space. - Show that X itself is a Baire space. T

A: Given: X is a topological space. To Prove: We have to show that the topological space X itself is a…

Q: Let f : X → Y be a function where X is a nonempty space and Y is a topological space with a given…

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Q: 3) Let R be the set of all real numbers with the euclidean topology and f: R → R be continuous…

A: Yes, the image of a continuous function from a path-connected space is always path-connected.…

Q: 5. Let X =[0,2). Define T = {[0,a):0<a<2}. %3D a) Show that T is a topology on X. b) What are the…

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Q: 5) Let A= (2,3) U {4} be a subset of R with the standard opology. Find Ə(A).

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Q: (i) Show that a subset A of a topological space X is closed if and only A = A (ii) Prove that A…

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Q: Exercise 2. Let X be a topological space. Endow Xx X with the product topology. Consider the map f:…

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Q: {X, Ø, {1}, {2},{1,2},{1,3}}, is (X, t) a connected a) Let X = {1,2,3} with the topology t=…

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Q: 5) Let X be a topological space. Is there always a metric on X which gives rise to its topology?…

A: No, there does not always exists a metric in a topological space that gives rise to its topology.…

Q: Let X={a,b,c} and Y={1,2,3} and t={Y,0,{1,2}}. The function f={(a,1),(b,3),(c,2)}. Then the topology…

A: X = { a, b, c } , Y = { 1, 2, 3 } T = { Y, ∅,  {1, 2} } f = { (a, 1), (b, 3), (c, 3) }

Q: Let (X, T) be a topological space and A C X. Prove that A =A.

A: PROOF :

Q: True or False: Consider the subsets (0, 1) and (2, 3) of R, the set of all real numbers with the…

A: The given statement is  Consider the subsets (0, 1) and (2, 3) of R, the set of all realnumbers with…

Q: Let (X, T1) and (Y, T2) be topological spaces. Show that the set {U × Y ∣ U ∈ T1} ∪ {X × V ∣ V ∈…

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Q: Theorem 2.3. A set U is open in a topological space (X,J) if and only if for every point x€ U, there…

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Q: Exercise 1. Let X be a metric space. The subsets of X which are arbitrary (finite or infinite)…

A: What is Open Set: Open sets in a metric space are those that, for every point P, include all other…

Q: Let V = F" where FE {C, R} show that V is a Banach space with the norm ||x| = maxisisn{xil VxEF".

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Q: let (2, m) be a meesurable space. then 1f f9: x LJare measural ble function, then -> ) f43 is mea…

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Q: Consider the topology T on X= (a, b, c, d, e), (X, Ø, (a), (e,d}, {a, e, d), (b, c, d, e}} Show that…

A: It is given that,  X = {a, b, c, d, e} and  T = X, ∅, a, c, d, a, c, d, b, c, d, e be a topology on…

Q: Exercise 2. Let X be a topological space. Endow X X X with the product topology. Consider the map f:…

A: We have to show for every closed subset C , f(C) is closed in X×X

Q: Theorem 2.22. Let A and B be subsets of a topological space X. Then: (1) A C B implies Ā c B. (2)…

A: Theorem 2.2 (i) Let A and B subsets of a topological space X such that A⊂B i.e. A is contained in B.…

Q: Q2. Answer as true or false: (a) f-'((AUB)nC) = (ƒ-'(A)n f-1C)) U (f-1(B) n f-1(C)). (b) Given the…

A: (a) is True. (b) is False.

Q: Let X be a set with more than one element. (X,t) is a topological space such tha Vxe X,{x} €T then…

A: Given a set X with more than one element.

Q: B) Let (R, T) be the left ray topological space If Y=(3,7], then define ty and determine the open…

A: Given: B) ℝ, τ is a left ray topological space and Y=(3, 7]. To determine: Which one of the open and…

Q: Prove that Cl(Q) = R in the standard topology on R. %3D

A: The closure of S may equivalently be defined as the union of S and its boundary, and also as the…

Q: Exercise 2. Let X be a topological space. Endow Xx X with the product topology. Consider the map f:…

A: We have to give an example of a topological space X such that the diagonal is not closed in X×X

Q: (a) Let X be a topological space. Let q: X→ A be a quotient map and let p: A → B be a surjection…

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Q: Let x be an infinite set equipped with the indiscrete topology T then (X,T) is compact. True or…

A: An infinite set equipped with discrete topology is not compact.